Abstract: We consider a stochastic fluid queue served by a constant rate server anddriven by a process which is the local time of a certain Markov process. Such astochastic system can be used as a model in a priority service system,especially when the time scales involved are fast. The input local time inour model is always singular with respect to the Lebesgue measure which in manyapplications is ``close- to reality. We first discuss how to rigorouslyconstruct the necessarily unique stationary version of the system under somenatural stability conditions. We then consider the distribution of performancesteady-state characteristics, namely, the buffer content, the idle period andthe busy period. These derivations are much based on the fact that the inverseof the local time of a Markov process is a L\-evy process a subordinatorhence making the theory of L\-evy processes applicable. Another importantingredient in our approach is the Palm calculus coming from the point processpoint of view.